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Let $\triangle ABC$ be a right triangle, with the point $H$ the foot of the altitude from $C$ to side $\overline{AB}$.

Prove that

Oct 29, 2018

#1
+3726
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Try to expand the terms!

Oct 30, 2018
#2
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Expanding the terms,

$$(x^2+2xh+h^2)+(y^2+2yh+y^2)=(a^2+2ab+b^2)$$

Using Pythagorean Theorem,

$$x^2+h^2=a^2\\ y^2+h^2=b^2$$

We could subsitute the values in, and rewrite the equation

$$a^2+2xh+b^2+2yh=a^2+2ab+b^2\\ 2xh+2yh=2ab\\ h(x+y)=ab$$

$$[ABC]=\frac12 AB\cdot CH = \frac12 BC \cdot AC\\ \frac12 (x+y)h=\frac12 ab\\ h(x+y)=ab$$

I hope this helped,

Gavin

Oct 30, 2018
edited by GYanggg  Oct 30, 2018
#3
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Note that triangle  BCA  is similar to triangle  BHC

Which implies that

HC / BC  =  CA / BA..... so...

h / a  =  b / ( x + y)    (1)

Now expand   ( x + h)^2 + ( y + h)^2  =  ( a + b)^2    (2)

x^2 + 2xh + h^2  + y^2 + 2yh + h^2  =  a^2 + 2ab + b^2     (3)

And since  x^2 + h^2  =  a^2    and   y^2 + h^2  = b^2

We can subtract these equal parts from  (3)  and we are left with

2xh + 2yh  =  2ab        divide through by 2

xh + yh  =  ab      factor out h on the left

h ( x + y)  =  ab       rearrange as

h / a  =  b / ( x + y)     but, by (1)....this is true

So (2)  must be true, as well

Oct 31, 2018